Determine the coefficient of 𝑥 to the power of negative six in the expansion of 𝑥 plus one over 𝑥 squared to the power of six.
In this question, we have a binomial expansion written in the form 𝑎 plus 𝑏 to the 𝑛th power. We know that the general term denoted 𝑎 sub 𝑟 plus one is equal to 𝑛 choose 𝑟 multiplied by 𝑎 to the power of 𝑛 minus 𝑟 multiplied by 𝑏 to the power of 𝑟. We will begin by rewriting our expression as 𝑥 plus 𝑥 to the power of negative two raised to the sixth power. We can do this as one over 𝑥 to the power of 𝑛 is equal to 𝑥 to the power of negative 𝑛. The general term of this expansion will therefore be equal to six choose 𝑟 multiplied by 𝑥 to the power of six minus 𝑟 multiplied by 𝑥 to the power of negative two to the power of 𝑟. This can be rewritten as six choose 𝑟 multiplied by 𝑥 to the power of six minus 𝑟 multiplied by 𝑥 to the power of negative two 𝑟.
We know from our laws of exponents that when the base is the same, we can add the exponents or powers. Our expression becomes six choose 𝑟 multiplied by 𝑥 to the power of six minus 𝑟 plus negative two 𝑟. Simplifying this again gives us six choose 𝑟 multiplied by 𝑥 to the power of six minus three 𝑟. In this question, we’re interested in the coefficient when the exponent of 𝑥 is negative six. We need to calculate the value of 𝑟 when six minus three 𝑟 equals negative six. We can subtract six from both sides of this equation so that negative three 𝑟 is equal to negative 12. Dividing both sides by negative three gives us 𝑟 is equal to four. We can now substitute this value back into the expression for our term.
This gives us six choose four multiplied by 𝑥 to the power of negative six. The coefficient of this term is therefore equal to six choose four. We know that 𝑛 choose 𝑟 or 𝑛C𝑟 is equal to 𝑛 factorial divided by 𝑛 minus 𝑟 factorial multiplied by 𝑟 factorial. Six choose four is therefore equal to six factorial divided by four factorial multiplied by two factorial. Six factorial is the same as six multiplied by five multiplied by four factorial. Canceling the four factorial gives us six multiplied by five divided by two factorial. As two factorial is equal to two and 30 divided by two is 15, six choose four is equal to 15. The coefficient of 𝑥 to the power of negative six in the expansion is 15.